After multiplying by 5, each of the following numbers will have the same number of perfect square factors EXCEPT

Options

A. 350
B. 290
C. 250
D. 12

Answer

To solve the problem, we need to determine how many perfect square factors each of the given numbers (350, 290, 250, and 12) has after being multiplied by 5. We will analyze each number step by step.

### Step 1: Factor each number

1. **For 350:**
- Prime factorization: \(350 = 5^2 \times 7^1 \times 2^1\)

2. **For 290:**
- Prime factorization: \(290 = 5^1 \times 29^1 \times 2^1\)

3. **For 250:**
- Prime factorization: \(250 = 5^3 \times 2^1\)

4. **For 12:**
- Prime factorization: \(12 = 2^2 \times 3^1\)

### Step 2: Multiply each number by 5

1. **For 350:**
- \(350 \times 5 = 1750\)
- Prime factorization: \(1750 = 5^3 \times 7^1 \times 2^1\)

2. **For 290:**
- \(290 \times 5 = 1450\)
- Prime factorization: \(1450 = 5^2 \times 29^1 \times 2^1\)

3. **For 250:**
- \(250 \times 5 = 1250\)
- Prime factorization: \(1250 = 5^4 \times 2^1\)

4. **For 12:**
- \(12 \times 5 = 60\)
- Prime factorization: \(60 = 5^1 \times 2^2 \times 3^1\)

### Step 3: Count the perfect square factors

To find the number of perfect square factors, we use the formula:
If a number has the prime factorization \(p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_n^{e_n}\), the number of perfect square factors is given by:
\[ \text{Number of perfect square factors} = (f_1 + 1)(f_2 + 1) \ldots (f_n + 1) \]
where \(f_i\) is the largest even integer less than or equal to \(e_i\).

1. **For 1750 (from 350):**
- \(5^3\) contributes \(1\) (since \(3\) is odd, we take \(2\))
- \(7^1\) contributes \(0\) (since \(1\) is odd)
- \(2^1\) contributes \(0\) (since \(1\) is odd)
- Total: \((1+1)(0+1)(0+1) = 2 \times 1 \times 1 = 2\)

2. **For 1450 (from 290):**
- \(5^2\) contributes \(1\) (since \(2\) is even)
- \(29^1\) contributes \(0\) (since \(1\) is odd)
- \(2^1\) contributes \(0\) (since \(1\) is odd)
- Total: \((1+1)(0+1)(0+1) = 2 \times 1 \times 1 = 2\)

3. **For 1250 (from 250):**
- \(5^4\) contributes \(2\) (since \(4\) is even)
- \(2^1\) contributes \(0\) (since \(1\) is odd)
- Total: \((2+1)(0+1) = 3 \times 1 = 3\)

4. **For 60 (from 12):**
- \(5^1\) contributes \(0\) (since \(1\) is odd)
- \(2^2\) contributes \(1\) (since \(2\) is even)
- \(3^1\) contributes \(0\) (since \(1\) is odd)
- Total: \((0+1)(1+1)(0+1) = 1 \times 2 \times 1 = 2\)

### Conclusion

- 350 has 2 perfect square factors.
- 290 has 2 perfect square factors.
- 250 has 3 perfect square factors.
- 12 has 2 perfect square factors.

The number that does not have the same number of perfect square factors after multiplying by 5 is **250**, as it has 3 perfect square factors while the others have 2.